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Interactions Among Mutualism, Competition, and Predation Foster Species Coexistence in Diverse Communities

Interactions Among Mutualism, Competition, and Predation Foster Species Coexistence in Diverse Communities

Theor Ecol DOI 10.1007/s12080-015-0251-2

ORIGINAL PAPER

Interactions among mutualism, , and foster coexistence in diverse communities

Benedicte Bachelot & María Uriarte & Krista McGuire

Received: 23 December 2014 /Accepted: 2 January 2015 # Springer Science+Business Media Dordrecht 2015

Abstract In natural systems, organisms are simultaneously deriving greater benefits from mycorrhizal associations than engaged in mutualistic, competitive, and predatory interac- rare species. tions. Theory predicts that species persistence and stability are feasible when the beneficial effects of mutualisms Keywords Prey–predator model . dynamics . are balanced by density-dependent negative feedbacks. Mutualism . Janzen-Connell . Mycorrhizal associations Enemy-mediated negative feedbacks can foster plant species coexistence in diverse communities, but empirical evidence remains mixed. Disparity between theoretical expectations and empirical results may arise from the effects of mutualistic mycorrhizal fungi. Here, we build a multiprey species/ Introduction predator model combined with a bidirectional ex- change system, which simulates mutualistic interactions be- Interspecific interactions play a key role in driving population tween and fungi. To reach population persistence, (1) dynamics (Gause and Witt 1935). A large number of theoret- the per capita rate of increase of all plant population must ical and empirical studies have focused on understanding the exceed the sum of the negative per capita effects of predation, effects of competition and predation on interspecific competition, and costs of mycorrhizal associa- (Holling 1959; Rosenzweig and MacArthur 1963; Tilman tion, and (2) the per capita numerical response of enemies to 1982; Chesson 2000; Chesson and Kuang 2008). More re- mycorrhizal plants must exceed the magnitude of the per cently, studies have also investigated the role of mutualisms capita enemy rate of mortality. These conditions reflect the in isolation (Bronstein 2001a; Holland et al. 2002;Bever balance between regulation and facilitation in the system. In- 2003; Bruno et al. 2003). Two generalizations can be drawn teractions between plant natural enemies and mycorrhizal fun- from mutualism studies (Holland et al. 2002): mutualisms gi lead to shifts in the strength and direction of net mycorrhizal involve costs and benefits for both partners that are likely effects on plants over time, with common plant species density-dependent (Roughgarden 1975; Addicott 1979;Mo- rales 2000;Bronstein2001b), and population stability re- quires that the positive feedbacks derived from mutualism are balanced by negative feedbacks (Gause and Witt 1935; Electronic supplementary material The online version of this article Vandermeer and Boucher 1978;Chesson2000; Bever 2003). (doi:10.1007/s12080-015-0251-2) contains supplementary material, which is available to authorized users. Stabilizing negative feedbacks can arise from multiple mechanisms such as resource limitation, competition, preda- * : B. Bachelot ( ) M. Uriarte tion, or change in the net effects of mutualism (i.e., the balance Department of , and Environmental , Columbia University, 1200 Amsterdam Avenue, New between its costs and benefits) as increases York, NY 10027, USA (Holland et al. 2002; Schmitt and Holbrook 2003; Holland e-mail: [email protected] and DeAngelis 2010; Holland et al. 2013). Recent theoretical studies have enhanced our understanding of the joint role of K. McGuire Department of Biology, Barnard College, Columbia University, New mutualisms and negative feedbacks in stabilizing species in- York, NY 10027, USA teractions; however, they often remain limited to one or two Theor Ecol aspects of interspecific interactions. In natural systems, organ- benefits in analysis of mutualism (Neuhauser and Fargione isms are simultaneously involved in mutualism, competition, 2004; Holland and DeAngelis 2010; Holland et al. 2013), and predation. Each process interacts with each other, limiting costs are usually lacking and mutualism is generally unidirec- our ability to understand their joint effects from studies that tional in theoretical studies that combine mutualism and an- investigate them in isolation (Fontaine et al. 2011;Georgelin tagonism (Holland et al. 2013; Georgelin and Loeuille 2014; and Loeuille 2014). Therefore, it is important to combine mu- Mougi and Kondoh 2014). tualism and food web dynamics into our understanding of The relative effect of mycorrhization on a plant is typi- community dynamics and species coexistence (Rai et al. cally gauged in terms of the quantity of limiting resources 1983; Addicott and Freedman 1984; Freedman et al. 1987; extracted from soil versus the amount of carbon needed to Ringel et al. 1996;Jang2002; Bronstein et al. 2003;Melián maintain the . Evidence for positive mycorrhizal et al. 2009;Loeuille2010; Mougi and Kondoh 2012;Holland effects on plants is extensive and has been attributed to et al. 2013; Georgelin and Loeuille 2014; Mougi and Kondoh a variety of mechanisms including greater plant nutrient 2014). Recently, there has been a regained interest in combin- uptake (Smith and Read 2008), defense against enemies ing interaction types such as mutualism and antagonism mo- (Gange and West 1994), and drought (Auge tivated by the recognition that indirect interactions can alter et al. 1987). The nutrient benefits a plant species derives the effects of mutualism and antagonism (Holland et al. 2013; from mycorrhizae increase with mycorrhizal fungal abun- Tang et al. 2014), affect resilience of dynamical systems to dance as colonization increases (Fitter 1991;Vander perturbations (Georgelin and Loeuille 2014), and promote sta- Heijden et al. 1998; Lekberg and Koide 2005;Hoeksema bility in complex systems (Mougi and Kondoh 2014), which et al. 2010; Zhang et al. 2011) and saturates when all avail- tend to be unstable when investigating interaction types in able have been colonized (Smith and Read 2008; isolation (Allesina and Tang 2012). Vannette and Hunter 2011;Fig.1b(1)). Studies have also In this study, we investigate the joint role of predation, uncovered negative feedbacks to plants from mycorrhizal competition, and mutualistic associations with arbuscular my- fungal associations, which result from greater carbon costs corrhizal fungi for the dynamics of multiple plant species and with higher fungal (Vannette and Hunter 2011). for species coexistence. In highly diverse plant communities, However, the cost of mycorrhizal associations to the plant such as tropical forests, negative density-dependent factors saturates with plant abundance (Leon and Tumpson 1975; (also known as Janzen-Connell (J-C) effects; Janzen 1970; Kiers et al. 2011;Fig.1b(2)). Connell 1971; Connell et al. 1984) are the most frequently Additionally, mycorrhization has been repeatedly dem- studied mechanisms that could explain the species coex- onstrated to minimize the risk of by root patho- istence (reviewed in Wright 2002). Janzen-Connell effects op- gens (Newsham et al. 1995) and to maximize response ef- erate through the attraction of species-specific enemies such as ficiency of plant to natural enemies (Bi et al. 2007). Con- predators, , or pathogens to the seedlings of sequently, increased mycorrhizal colonization should be canopy , which reduce conspecific seedling survivorship correlated with decreased vulnerability to enemies, poten- near the adult trees at high conspecific seedling density, leav- tially leading to weaker negative enemy-mediated density ing ecological space for heterospecific seedlings to recruit. dependence and lower number of natural enemies The prediction from the J-C hypothesis is that the per capita (Fig. 1a(3)).Bietal.(2007) provide an extensive review predation rate of a plant species increases with plant species of the mechanisms underlying this effect, which include abundance at local and community scales (Fig. 1a(1)). increased production of allelochemicals such as phenols In contrast to the large body of research on the importance by the plants, induction of plant defense genes, activation of negative feedbacks for species coexistence in plant com- of the jasmonate pathway that is crucial to respond to at- munities and of positive feedbacks for plant invasion tacks by plant enemies, and attraction of predators of plant (Eppstein et al. 2006;Levineetal.2006; Eppstein and natural enemies (Hoffman et al. 2011a, b). Molofsky 2007), the role of fluctuating positive and negative Here, we present a mathematical model to explore the mycorrhizal feedbacks on plant community dynamics remains joint effects of negative enemy and mycorrhizal fungal underexplored both theoretically (Bruno et al. 2003) and em- feedbacks on plant community dynamics and species coex- pirically (Booth and Hoeksema 2010). Yet, theoretical studies istence. We then interpret the results of our model in light of have shown the importance of fluctuating mutualisms in sta- existing empirical work on several aspects of the feedbacks bilizing interspecific interactions (Holland et al. 2013). Unlike we examine. Specifically, we address the following soil pathogens, which only have negative effects on their host questions: trees, mycorrhizal fungi can have variable impacts on their host plant along a continuum from mutualism to 1. How does the relative strength of negative feedbacks (Fig. 1a(2), Johnson et al. 1997; Johnson and Graham 2013). from natural enemies and the magnitude and direction of If recent theoretical studies have incorporated costs and mycorrhizal effects vary over time within a plant species Theor Ecol

Fig. 1 Conceptual model showing the expected relationships between population levels (Selosse et al. 2006; van der Heijden and Horton natural enemy, mycorrhizal fungi, and plant population dynamics. a 1 Per 2009; Kiers et al. 2011). 3 Per capita carbon benefits the fungi gain from capita predation from enemies increases with plant abundance and the plants increase and saturate with the plant abundance because mycor- saturates. We showed two possible forms in plain and dashed lines. 2 rhizal associations are limited by root surface. 4 Per capita costs of the The effects of the fungi on the plants are both positive (nutrients mycorrhizal associations to the fungi increase with plant abundance but provisioning) and negative (carbon costs). Similarly, the effects of the decrease with the abundance of the fungi. Fungi can regulate this cost by plants on the fungi are both positive (carbon gain) and negative controlling mycelial growth and the extent of the common mycelial net- (nutrients sink). These costs and benefits are further developed in b. 3 work (Selosse et al. 2006; van der Heijden and Horton 2009; Kiers et al. Per capita protection plants derive from mycorrhizal fungi linearly 2011). Recent evolutionary stable strategies analysis showed that regula- increases with the abundance of mycorrhizal fungi. b 1 Per capita tion of costs and benefits of mutualistic association at the individual level nutrient benefits provided to the plants by the fungi increase (Fitter leads to a cost and benefit control at the population level (Holland et al. 1991; Van der Heijden et al. 1998; Lekberg and Koide 2005; Hoeksema 2004). Therefore, fungal populations control the costs of being associated et al. 2010; Zhang et al. 2011) and saturate with fungal abundance be- with plants. c These interactions could result in the following patterns of cause mycorrhizal associations are limited by root surface and fungi con- population dynamics: at low density, a plant population experiences a trol the amount of nutrients delivered to the plant (Smith and Read 2008; predation release that allows it to increase in density despite low benefits Vannette and Hunter 2011). 2 Per capita carbon costs of mycorrhizal from mycorrhizal fungi. At a specific flipping point (the black dot), the associations to the plant increase with the abundance of the fungi predation and the costs of supporting the mycorrhizal associations are not (Vannette and Hunter 2011) but decrease with the abundance of the plant offset by the high mycorrhizal benefits, and the plant population density that can regulate the costs at the individual (Leon and Tumpson 1975; decreases. It is important to note that the net mycorrhizal effect (difference Tilman 1982;Koide1991; Johnson et al. 1997; Smith and Read 2008; between benefits and costs) will be species-specific and in part controlled Johnson and Graham 2013; Koide 1991; Salzer et al. 1997)and by the ratio of fungi and plant abundance

and what are the impacts of these fluctuations on the spe- Model formulation cies’ abundance? 2. How does the relative strength of negative feedbacks and Our model assumes that plant species abundance fluctuates mycorrhizal effects on rare and common plant species vary under pressure from three forces: the negative effects of over time and what are the impacts of these fluctuations on inter-specific plant competition (Xi), the negative impacts of plant species coexistence? one natural enemies’ population (Y) (i.e., pathogens and her- 3. How do mycorrhizae alter the interactions between bivores), and the effect of the mycorrhizal fungal population plants and natural enemies? Similarly, how do natural en- (M). Mycorrhizal fungi and plants interact according to a bi- emies affect the mutualistic interaction between plants and directional resource exchange system; the fungi provide the mycorrhizal fungi? plants with increasing nutrients, whereas the plants provide Theor Ecol the fungi with carbon (Smith and Read 2008). Furthermore, couple a multispecies, continuous, nonlinear prey–predator mycorrhizal colonization lowers predation from plant enemies model (enemies-plant) with a bidirectional resource exchange by increasing plant defenses (Pozo and Azcón-Aguilar 2007; model to represent the interactions between plants and fungi.

Bi et al. 2007). To understand the dynamics that would result The dynamics of plant species i (Xi)attimet are described by from these complex enemy–plant–fungal interactions, we the following equation:

"# ðÞ Xn dX i t ¼ ðÞ þ α ðÞ; −β ðÞ; −½ŠðÞ; − ðÞ ðÞ− ðÞ ð Þ X i t bi iGi RM→X i t iLi RX i→M t Ci X t miMt Yt ai; jX j t 1 dt j¼1

where Xi(t)andY(t) represent population abundance of plant capita protection (mi) against enemies (Fig. 1a(3); Gange and species i (prey) and enemies, respectively, bi represents the per West 1994; Pozo and Azcón-Aguilar 2007;Bietal.2007). In capita exponential rate of increase of plant species i, ai,j des- order to prevent that a net positive effect arises from the inter- ignates the per capita interspecific competition among plant actions between natural enemies and fungi, we bounded the species, ai,i represents intraspecific competition (self-limita- mycorrhizal fungal abundance so that Ci(X,t)−miM(t)≥0. Fi- tion), M(t) is the abundance of the mycorrhizal fungal popu- nally, the plant nutritionally benefits (Fig. 1a(2), b(1), α ðÞ; lation, XiCi(X,t) is the functional response of natural enemies iGi RM→X i t ) and pays a carbon cost (Fig. 1a(2), b(2), β ðÞ; to plant species i, and therefore Ci(X,t)isthepercapitarateof iLi RX i→M t ) from the mycorrhizal associations (Smith mortality from one enemy (Fig. 1a(1)). We used a classical and Read 2008; Johnson and Graham 2013). The functions

Holling type III functional response (Holling 1959; Smout Gi(RM−>Xi, t) represent the per capita increase in resource et al. 2010): plants’ gain from the mycorrhizal associations (Fig. 1b(1);

Smith and Read 2008), whereas Li(RXi−>M, t) represents the ciX iðÞt X iCiðÞ¼X ; t X iðÞt X ; withallci andwi > 0 per capita carbon losses to the associations (Fig. 1b(2); Olsson 1 þ w c X ðÞt 2 et al. 2010). The parameter α represents the conversion of the j j j j i gain in a unit of resource per capita to a gain in abundance of ð2Þ the population of the plant species i.Incontrast,βi stands for the conversion of per capita loss in carbon to a loss in abun- dance of the population of the plant species i.Weusetheterm where wi is the handling time of plant species i,andci is the net mycorrhizal effect for the plant species i to denote the encounter rate of natural enemy with plant species i. This difference between the benefits (nutrition and protection from enables us to evaluate natural enemy behavior that causes enemies) and the costs (carbon): αi Gi(RM−>Xi, t)+miM(t)Y(t) density-dependent mortality (Janzen 1970; Connell 1971; and βi Li(RXi−>M, t). Murdoch 1975; Pacala and Crawley 1992;Mordecai2011), The dynamics of the mycorrhizal fungal population while acknowledging that natural enemies might satiate at (M) directly depend on the balance between the carbon high plant density (Silvertown 1980). Furthermore, it intro- gains and nutrient losses they derive from plant species duces the notion that natural enemy behavior might not be 1ton (Koide and Elliott 1989; Fitter 1991). In an effort to influenced by very low plant density (Real 1977;Krebs increase model tractability, we did not account for differ- 1974; Jeschke et al. 2002). In a diverse community, very rare ent mycorrhizal fungal species. Growing evidence has plant species might escape predation because natural enemies shown that the specific community of mycorrhizal fungi might not find them (Jaenike 1990; Pacala and Crawley 1992; associated with a plant rather than the identity of individ- Hierro et al. 2005) or not target them due to change in diet/host ual mycorrhizal fungi controls the outcomes of the asso- breadth with plant availability (Fox and Morrow 1981;Ste- ciations (Johnson and Graham 2013;Tojuetal.2013). phens and Krebs 1986). Therefore, a Holling type III func- Therefore, considering the mycorrhizal fungal community tional response is the best choice to investigate the joint effect rather than individual mycorrhizal fungal species does not of natural enemies and mycorrhizal fungi on plant dynamics. dramatically hinder the realism of our model. The dynam- Although we will present results for a type III response here, ics of the mycorrhizal fungal population are described by more general assumptions about functional responses yield a –resource (C-R) model (Rosenzweig and similar qualitative results (see Appendix 1). The term mi× MacArthur 1963) similar to the bidirectional C-R model M(t)inEq.1 modifies the functional response of the enemy proposed by Holland and DeAngelis (2010) as follows because mycorrhizal fungi provide plants with increased per (Fig. 1a(2)): Theor Ecol

! ðÞ Xn ÂÃspecies i and for fungal population M, respectively. Similarly, dM t ¼ ðÞ α ðÞ; −β ðÞ; − Mt M;iGM;i RX i→M t M;iLM;i RM→X i t rM li and lM,i are the saturation levels of the loss of resources, and dt i¼1 k and k are the half saturation constants. Parameters g and l ð3Þ i M,i represent the per capita interaction strength of one species on the other. We assumed per capita resource gain to plant species where GM,i (RXi−>M, t)andLM,i (RM−>Xi, t), respectively, rep- resent the per capita resource gains and losses of the fungal i (the Gi function in Eq. 1) a saturating (Michaelis-Menten) function of the abundance of fungi associated with each plant population to plant species i, αM,i represents the conversion of the gain in resource from plant species i to fungal population species i=1 to n (Fig. 1b(1)). Similarly, benefits to the mycor- rhizal fungi (the GM,i function in Eq. 3) are described by abundance (Smith and Read 2008), and βM,i modulates the per capita loss in resource to plant species i in loss in fungal Michaelis-Menten functions of the plant abundance associat- ed with the fungal population (Fig. 1b(3)). This functional abundance (respectively Fig. 1b(3, 4), and rM is the per capita exponential rate of mortality of the mycorrhizal fungi in the form captures the assumption that the benefits extracted by absence of plants. fungi and plants are limited by the extent of fine root surface Studies have shown that the effects of mycorrhizal associ- area and that the two symbiotic partners (i.e., plants and fungi) ations are context-specific, and the same mycorrhizal fungal can control these benefits. Per capita resources lost by plant species could have positive, negative, and neutral effects species i (the Li function in Eq. 1, Fig. 1b(2)) increase linearly (Smith and Read 2008; Johnson and Graham 2013;Toju with increasing fungal populations (Eq. 2), while resources et al. 2013). To account for this level of variation in the effect lost by the fungi (the LM,i functions in Eq. 3) increase linearly of mycorrhizal fungi, we use plant species-specific resource with greater abundance of plant species i population (Eq. 7, gain and loss functions (G and L in Eqs. 1 and 3) and plant Fig. 1b(4)). The per capita loss of resources functions saturates species-specific numerical responses (α and β in Eqs. 1 and with the supplier species abundance (denominators in Eqs. 5 3). The mycorrhizal fungi are strictly obligate and cannot sur- and 7, Fig. 1b(2, 4)). In Eq. 5 and Fig. 1b(2), the rationale for vive without the plants (Smith and Read 2008). Therefore, the the denominator is that plants have the potential to control dynamics of the fungi are solely controlled by the exchange of fungal colonization to maximize the benefits of the mycorrhi- resources with the plants. zal associations while minimizing the costs (Koide 1991; Quantifying the costs and benefits plants derive from Johnson et al. 1997; Smith and Read 2008; Johnson and Gra- mycorrhizal associations and vice versa remains challeng- ham 2013). Similarly, the rational for the denominator in Eq. 7 ing (Johnson et al. 1997; Johnson and Graham 2013). In our and Fig. 1b(4) is that fungi are able to overcome plant controls model, we used classical resource exchange functions such of the colonization (Johnson et al. 1997; Smith and Read as the one described by Holland and DeAngelis (2010) 2008; Johnson and Graham. 2013). In other words, plants (Fig. 1b(1, 2)): and fungi can control the costs of the mycorrhizal associations at the individual level and thus at the population level due to ðÞ ðÞ¼; giMt ð Þ evolutionary stable strategies of mutualism (Holland et al. Gi RM→X i t 4 hi þ MtðÞ 2004). Another motivation behind the denominator of the cost function arises from the fact that plants and mycorrhizal fungi are connected via common mycelial network, allowing them to share the costs of the mycorrhizal association with each liMtðÞ L ðÞ¼R → ; t ð5Þ other (Simard and Durall 2004). We acknowledge that the i X i M k þ X ðÞt i i form we are using to represent the mycorrhizal effect is a simplification because we do not know the exact forms of

In parallel, GM,i and LM,i for the fungi in Eq. 3 take the form per capita resource gains (the Gi and GM,i functions) and losses (Fig. 1b(3, 4)): (the Li and LM,i functions). These simple forms of consumer– resource interactions have been shown to be accurate in some ðÞ gM;iX i t G ; ðÞ¼R → ; t ð6Þ biological systems (Holland 2002; Holland et al. 2002;Hol- M i X i M þ ðÞ hM;i X i t land and DeAngelis 2006; Holland and DeAngelis 2010;Hol- land et al. 2013). Finally, enemy dynamics (Y) are described as one popula- ðÞ tion of enemies exhibiting host preferences rather than spe- ðÞ¼; lM;iX i t ð Þ LM;i RM→X i t 7 cialization. We assume that predation increases with plant kM;i þ MtðÞ abundance (Fig. 1a(1)). The assumption of host preferences can lead to the simulation of inappropriate population dynam- where gi and gM,i are the saturation levels of resource gains, ics for the few plants that have species-specific interactions and hi and hM,i are the half saturation constants for plant with specialist pathogens and herbivores. However, in these Theor Ecol associations between plants and specialist enemies, plant mutualistic interactions and food web dynamics. Holland abundance dynamics are likely to be tightly dependent on et al. (2013) used resource exchange functions to investigate the dynamics of their own specialist enemies and less depen- the stability of a system that included one plant species, one dent on other plant species present in the community. There- mutualistic , and one -parasite. In their system, fore, ignoring the distinction between host-specific and gen- the mutualistic pollinator gains benefits from the plant without eralist interactions in the system should not affect the validity paying a cost, and the nectar-parasite does not alter the behav- of our conclusions especially given the low rates of host spe- ior of the mutualistic pollinator (unidirectional mutualism). cialization of natural enemies in the tropics (Novotny et al. Our study expands on Holland et al. system (2013) and aims 2010). The dynamics of plant natural enemies are a function of to explore community dynamics and coexistence of n plant plant species and mycorrhizal fungal abundances as follows: species that are involved in bidirectional exchange with my- "#corrhizal fungi, where each plant and experiences ben- Xn dYðÞ t efits and costs. Furthermore, natural enemies whose functional ¼ YtðÞ ½Šd ðÞC ðÞX ; t −m MtðÞX ðÞt −eYðÞ t −r i i i i response is affected by mycorrhizae target the n plant species. dt i¼1 To evaluate the system, we assume that the mycorrhizal ð8Þ fungal population does not tend towards infinity but rather exhibits an upper bound (v). We defined this upper bound to ensure that the protection conveyed by mycorrhizal fungi does where di Ci (X,t)Xi(t) represents the per capita numerical re- not result in a positive effect of natural enemies on plant pop- l l sponse of enemies to plant i, mi the per capita protection Ci Ci ulation (for all plant species i, mi ≤ def ). In other against enemies provided by the mycorrhizal associations maxðÞMtðÞ v with plant species i (Fig. 1a(3)), M the abundance of the my- words, the mycorrhizal fungi might enhance plant protection corrhizal fungal population, e is the per capita competition against natural enemies, but this enhancement will not result coefficient for the enemy species (self-limitation), and r the in a total protection. If the mycorrhizal fungal population goes per capita exponential rate of mortality of the enemy species in to , then our model becomes a classical multispecies – the absence of plant (i.e., starvation rate). Therefore, the my- prey predator type system. Therefore, in our work, we inves- corrhizal fungi have two effects on the plant natural enemy tigate the conditions to maintain plant species coexistence population (Wootton 1994; Strauss and Irwin 2004; when the mycorrhizal fungal population remains between a Koricheva et al. 2009; Bardgett and Wardle 2010; Barber strictly positive lower bound ( ) and an upper bound (v). The et al. 2012): An indirect effect that arises from modifying plant model was evaluated for n plant species (Appendix 1). Spe- population growth (depicted in the resource exchange func- cifically, we analyzed the model to find conditions allowing tions, G and L), and a direct effect that emerges from change in the populations to remain between a positive lower abundance natural enemy functional response due to increased plant pro- boundary and a finite upper abundance boundary. In other words, we evaluate the conditions enabling the coexistence tection (−miM). We chose to co-model community dynamics in continuous of n plant species in the system. We also analyzed a simple time, which is more suitable to model populations of organ- version of the model involving one plant species and a fixed isms with short life spans (natural enemies and mycorrhizal mycorrhizal fungal population in order to understand how fungi) and very long-lived ones (trees, Wangersky 1978). The mycorrhizal fungi alter the prey–predator dynamics final system is a nonlinear prey–predator model that combines (Appendix 2). Similarly, we investigated the case of a fixed competition among plant species, predation, and potential fa- natural enemy population to understand its effect on the mu- cilitation by mycorrhizal fungi if the overall net mycorrhizal tualistic interactions (Appendix 2). We illustrated the impor- effect is positive. This last effect is a fluctuation-dependent tant results of the model using simulation to represent fluctu- mechanism (Chesson 2000). Over time, as plant abundance ations in abundance over time within one plant species and for increases, the facilitation effect increases until a flipping point two coexisting plant species: one rare and one common. In after which this facilitation effect levels off, eventually becom- this paper, the distinction between a rare and a common plant ing a negative density-dependent effect driven by a growing species is a relative notion based upon the relative density of enemy population and the carbon and other costs associated each plant species. Finally, we investigated the dynamics of with the mycorrhizal symbiosis (Fig. 1c). The formulation of one equilibrium by using simulations and changing the value the net mycorrhizal effect via bidirectional resource/consumer of each parameter one by one (Tables 2 and 3 in Appendix 3). system (Johnson et al. 1997; Holland and DeAngelis 2010) Specifically, we wanted to assess how small changes in each enables this effect to become negative and introduces an ad- parameter of the model with two plant populations would ditional negative covariance between and density sim- change the dynamics of the equilibrium (stable, periodic, not ilar to the one produced by the effect of predation (Chesson periodic, or unstable). All simulations were accomplished 2000). Our model is one of the first attempts to combine using Mathematica v. 7.0. (Wolfram Research, Inc. 2008). Theor Ecol

Results and discussion Temporal fluctuations in negative feedbacks and the strength and direction of the net mycorrhizal effect correlate Variability of the relative strength of negative feedbacks with fluctuations in the abundance of the plant and enemy and net mycorrhizal effect on plant species populations. The fluctuations in the abundance of the plant population can be stable and stationary or periodic in the case Our model shows that when a plant species reaches high abun- of periodic or constant parameters (for example parameters that dance, its decline can occur via three mechanisms. First, at fluctuate seasonally) (step 3 in Appendix 1). The simultaneous high density, it suffers greater per capita mortality from enemy analysis of the dynamics of plants, enemies, and mycorrhizal

(Y(t)Ci(X,t)inEq.1) than at low density. Second, intraspecific fungi provides insights into the mechanisms underlying plant competition (ai,iXi(t)inEq.1)isstrongerathighrelativeto population dynamics. If the per capita benefits from mycorrhi- low conspecific abundance. Third, the positive effect from zal fungi to the plant is high relative to the per capita costs ðÞ ðÞþα ðÞ; ðÞ ðÞþα ðÞ; > β ðÞ; mycorrhizal associations (miMtYt iGi RM→X i t ) (miMtYt iGi RM→X i t iLi RX i→M t ), the dy- does not compensate for the negative effect of natural enemies namics of the net mycorrhizal effect and plant populations will as the mycorrhizal associations become more costly in term of be positively correlated and the net mycorrhizal effect and en- carbon. These three factors then lead to a decline in plant emy population will be negatively correlated (Fig. 2a). In other abundance. The opposite scenario applies when a plant spe- words, mycorrhizal fungi will have a large impact of the dy- cies reaches low abundance. namics of plants and enemies by enhancing plant population

Fig. 2 Model simulations for one plant species (X) with constant dynamics. Panel b represents the opposite situation where the parameters in the presence of natural enemies (Y) and mycorrhizal costs of the mycorrhizae are greater than its benefits ðÞðÞ ðÞþα ðÞ; < β ðÞ; fungi (M). M1 represents the net effect of mycorrhizal associations miMtYt 1G1 RM→X 1 t 1L1 RX 1→M t .Inthiscase, for the plant species. Panel a represents a situation where the plant and mycorrhizal fungal dynamics are synchronous. benefits of the mycorrhizal symbiosis are greater than the costs Parameter values used for the simulations are provided in ðÞðÞ ðÞþα ðÞ; > β ðÞ; miMtYt 1G1 RM→X 1 t 1L1 RX 1→M t .Inthiscase, Appendix 2 there is a mismatch between plant and mycorrhizal fungal Theor Ecol growth and impeding predation from enemies. Conversely, if mycorrhizal effect and the natural enemy population size the per capita benefits from mycorrhizal associations are lower (Fig. 3). than its per capita costs for both fungi and plant populations Our model shows that the joint impacts of enemies and ðÞ ðÞþα ðÞ; < β ðÞ; ( miMtYt iGi RM→X i t iLi RX i→M t andmiM mycorrhizal fungi can lead to temporal fluctuations in the ðÞ ðÞ þα ðÞ; < β ðÞ; t Yt M;iGM;i RX i→M t M;iLM;i RM→X i t ), the abundance of a single plant species. The impacts of these net mycorrhizal effect to the plants and the plant population combined feedbacks on plant dynamics have received scant size will remain positively correlated, but the net mycorrhizal attention (Van der Putten et al. 2009; Bardgett and Wardle effect to the plants and the enemy population might become 2010). While these interactions are complex and likely to be positively correlated (Fig. 2b). In other words, the mycorrhizal species-specific (Wardle 2002), evidence from herbivory stud- fungi will not be beneficial to the plants but rather they might ies suggests that mycorrhizal fungi and plant natural enemies indirectly enhance the enemy population. In other words, this are interacting (Bi et al. 2007). The various dynamic patterns could occur if by enhancing plant growth, mycorrhizal fungi obtained with our model suggest that the interactions between indirectly benefit plant enemies. If the net mycorrhizal effect plant natural enemies and mycorrhizal fungi are often but not fluctuates between positive and negative (The sign of miMtðÞ always negative. In support of our findings, a number of em- ðÞþα ðÞ; −β ðÞ; Yt iGi RM→X i t iLi RX i→M t will change over time pirical studies have uncovered negative correlations between due to fluctuating plant and fungal populations, similar to the herbivory and the abundance, phenology, and colonization of parasitism/mutualism continuum proposed by Johnson and mycorrhizal fungi, lending support to the idea that top-down Graham 2013), the dynamics will exhibit a variety of patterns effects can modulate the strength and direction of mycorrhizal ranging from positive to negative correlations between the net effects on plants. For example, Gehring and Whitham (1994)

Fig. 3 Model simulations for one plant species (X) in the presence of effect of mycorrhizal associations is fluctuating between positive and natural enemies (Y) and mycorrhizal fungi (M) with constant parameters. negative. Parameter values used for the simulations are provided in M1 represents the net effect of mycorrhizal associations for the plant Appendix 2 species. Panels a and b illustrate two situations where over time the net Theor Ecol found a decrease in mycorrhizal colonization following defo- of the maximum value of the per capita numerical responses liation for 23 of 37 studied plants. Recent reviews, however, n ∑ u of enemies to plants ( diCi pi)(step1inAppendix 1): highlight how the complex interaction between plant enemies i¼1 and mycorrhizal fungi can vary with the frequency, timing, n and type of herbivory (Klironomos et al. 2004; Saravesi X r < d Cup ð10Þ et al. 2008). i i i i¼1

Impact of negative feedbacks relative strength and net mycorrhizal effect fluctuations on species abundance This condition ensures the existence of an upper bound (q) and coexistence for the enemy population that depends only on the per capita n ∑ u enemy numerical responses ( diCi pi), the per capita com- Our model predicts coexistence of n plant species under the i¼1 joint effects of mycorrhizal fungi, plant natural enemies, and petition among enemies (e), and the per capita enemy expo- plant competition, which together establish lower and upper nential rate of mortality (r), as follows: abundance constraints, offering an additional mechanism to X n u − diCi pi r explain the high diversity of plant species in diverse commu- YtðÞ≤ i¼1 defq ð11Þ nities such as tropical forests. This result is the first theoretical e evidence showing that the joint effects of plant natural ene- mies and mycorrhizal fungi can lead to plant species coexis- tence. This result is consistent with theoretical work showing A third condition required to prevent enemy population that mutualism alone cannot lead to species coexistence extinction is that the total of the minimum value of the per n ÀÁ (McGill 2005) unless mutualistic populations are regulated ∑ le ; capita numerical responses of enemies to plants ( diCipi by other mechanisms (e.g., plant natural enemies in our mod- i¼1 e el, Simonsen and Stinchcombe 2014). where pi is the lower limit of plant species i population size) n Our model predicts that each plant species will fluctuate ∑ ðÞ− ; protected by mycorrhizal fungi ( dimivpi where v is the between low and high abundance constraints that depend on i¼1 the magnitude of negative feedbacks from natural enemies, upper bound of mycorrhizal fungal population) must be great- plant, and mycorrhizal fungi. In order to reach coexistence, er than the per capita enemy exponential rate of mortality (r) the parameters of the model must satisfy three conditions (Ap- (step 2 in Appendix 1): pendix 1). First, the per capita exponential rate of growth of Xn  the plant population i (bi) must be greater than the sum of (1) d Clep −d m vp > r ð12Þ u i i i i i i the negative per capita effects due to predation (Ci q,whereq i¼1 u is the upper bound of enemy population and Ci is the maximum value of the per capita mortality of plant species i from enemy, see Appendixhi 1), (2) the per capita interspecific If this condition is met (Eq. 12), then the condition to pre- n ∑ vent enemies from becoming overdominant (Eq. 10) is also competition ( ai; jp j ,wherepj is the upper limit of plant j¼1; j≠i met. Thus, we found that two necessary conditions must be species j population size), and (3) the per capita costs of my- met to maintain species coexistence: (1) the per capita expo- corrhizal associations (β liv, where v is the upper bound of nential rate of growth of the plant population must be greater i ki mycorrhizal fungal population) (step 2), that is: than the sum of the negative effects due to per capita preda- tion, per capita interspecific competition, and the per capita Xn hi u liv costs of mycorrhizal associations, and (2) the total of the min- b > C q þ a ; p þ β ð9Þ i i i j j i imum value of the per capita numerical responses of enemies j¼1; j≠i ki to plants protected by mycorrhizal fungi must be greater than the per capita enemy exponential rate of mortality. In other This first condition ensures that plant populations avoid words, the protection conveyed by the mycorrhizal associa- extinction. Second, the persistence of plant species requires tion is not strong enough to dramatically decrease the enemy that the enemy population does not become overdominant and population (food web cascade; Bruno et al. 2003;Cantrelland that this upper limit of natural enemy population is biological- Cosner 2001; McGill 2005). Overall, these two conditions ly reachable (in other words, it is not an infinite or a negative reflect the balance between regulation and facilitation in the number). This condition requires that the per capita enemy system. More empirical and experimental work is required to exponential rate of mortality (r) is strictly lower than the total evaluate the realism of these two conditions. Theor Ecol

If the two conditions described in inequalities Eqs. 9 and 12 species (αrgr<αcgc) as expected in light of empirical work are met, the abundance of plant species i (Xi) will fluctuate (Moora et al. 2004), given similar per capita intraspecific com- e between the following upper (pi) and lower (pi)limits(see petition (ai,i)andpercapitagrowthrate(bi), the maximum Appendix 1 for more details): abundance of a rare plant species (pr) will remain lower than that of a common plant species (pc): hi X n liv u b −β − a ; p −C q i i j¼1; j≠i i j j i b þ α g þ α þ α e ki ≤ ≤ i i i br rgr bc cgc pidef X i defpi p ¼ ≤ ¼ p ð14Þ ; ; r c ai i ai i ar;r ac;c ð13Þ

The parameters q and v represent the upper boundary of Therefore, inequality Eq. 14 predicts that lower frequency enemy and mycorrhizal fungal abundance, respectively (Ap- of association with beneficial fungi could explain the relative pendix 1). The upper limit of plant abundance depends on the rarity of a plant species. Alternatively, lower growth rate or higher intraspecific competition in rare relative to common per capita benefits to the plant species i (αigi)associatedwith the mycorrhizal associations and the per capita exponential plant species could explain rarity. The lower limit of plant abundance can take a lower value rate of increase of plant species i (bi), whereas the lower boundary is a function of the maximum per capita costs of for common species than for rare species when the effects of u β liv per capita predation (Ccq) plus the per capita costs of mycor- mycorrhizal associations to the plant species i ( i k ), the per i β liv rhizal associations ( i k ) are greater in a common relative to a capita competition among plants (aij), and the maximum per i u rare plant species (β lcv þ Cuq > β lrv þ Cuq), given similar capita predation effect (Ci ). Different parameters will lead to c kc c r kr r different model outputs: If the per capita benefits gained via per capita intraspecific and interspecific competition (ai,j)and mycorrhizal fungi are lower in rare relative to common plant percapitagrowthrate(bi):

hi hi X n X n lcv u lrv u b −β − a ; p −C q b −β − a ; p −C q c c j¼1; j≠c c j j c r r j¼1; j≠r r j j r e ¼ kc ≤ kr ¼ e ð Þ pc pr 15 ac;c ar;r Greater effect of natural enemies on common relative to other species. Future work should investigate how network rare plant species is predicted by the J-C hypothesis (Janzen structure would affect natural enemies–plants–mycorrhizal 1970;Connell1971; Connell et al. 1984). Furthermore, sev- fungi interactions. eral empirical studies supported this prediction (Bachelot and Kobe 2013; Bagchi et al. 2014). However, if plant natural Impacts of mycorrhizal associations on a plant-enemy u enemies equally target common and rare plant species (Ccq= dynamics u Crq), then the costs that arise from mycorrhizal associations could explain rarity (if rare species pay greater per capita costs Consistent with recent work (Holland et al. 2013;Georgelin for the association than common species) or large fluctuations and Loeuille 2014), the third analysis of the system showed in common plant species abundance (if common plant species that adding mutualism to a prey–predator system could stabi- experiences greater per capita costs than rare plant species). lize or destabilize the interactions. Specifically, our analysis Overall, variation in the strength and direction of enemy revealed complex changes in the dynamics of the system with and mycorrhizal fungal feedbacks confines the population of increasing mycorrhizal fungal abundance (Appendix 2). If the each plant species between an upper limit pi and a lower limit protection conveyed by the fungi to the plants is small enough, e pi, ensuring plant species coexistence because no species will mutualism stabilizes the interactions between plants and nat- go extinct or become overdominant. Furthermore, common ural enemies, suggesting that mutualism can damper the inter- plant species are likely to reach higher abundance and can actions between plants and natural enemies (Figs. 5 and 6). experience greater fluctuations in abundance over time than However, as mycorrhizal fungal abundance continues to in- rare plant species (Fig. 4). crease, the natural enemy population can be pushed to extinc- Beside change in the strength of various interactions tion, resulting in the collapse of the system (Fig. 6). When among species, the hierarchical structure of the interactions protection conveyed by mycorrhizal fungi to plants is large, can greatly impact species coexistence (Kondoh et al. 2010; mutualism destabilizes the system, which can introduce cycle Mougi and Kondoh 2012; Suweis et al. 2014; Mougi and dynamics and collapse (Fig. 6). Similar potential stabilization Kondoh 2014). In our model, however, we did not incorporate and destabilization effects of mutualism were also detected in hierarchical structure since each species interact with every pollinator––plant systems involving unidirectional Theor Ecol

Fig. 4 Model simulations for two plant species (X1 and X2). The dynamics of the dominant plant species and associated net mycorrhizal effect are shown in orange and dark blue,rarespecies in purple and light blue. Fluctuations in enemy (green) and mycorrhizal (black)effects lead to sporadic changes in the abundance status of the two plant species, with periods when the common plant species becomes less abundant than the rare plant species (area shaded in). M1 and M2 represent the net effect of mycorrhizal associations for the plant species X1 and X2. Parameter values used for the simulations are provided in Appendix 1

mutualism (Mougi and Kondoh 2012; Georgelin and Loeuille competition) could lead to the stability of many interacting 2014). species. They found that a moderate mixing of the three types Overall, stabilization was more likely at intermediate my- of interactions enhanced stability in diverse systems. Howev- corrhizal fungal population, which is somewhat consistent er, they assume functional type I interactions between each with a recent study of “hybrid” community (Mougi and species simplifying dramatically the system. Kondoh 2014). In this study, the authors asked which compo- Interestingly, the costs and benefits of mycorrhizal fungal sition of interactions (mutualism, antagonism, and associations have a similar effect on the Jacobian matrix, sug- gesting that both costs and benefits could stabilize prey–pred- ator interactions (Appendix 2). However, the upper and lower boundaries of the population abundance set some limits to the costs and benefits. If benefits tend to infinity, the system is destabilized because plant population obtains an infinite growth rate. In contrast, if costs tend to infinity, the plant population dies off. Therefore, within the limit of coexistence, increasing costs and benefits tend to stabilize the systems but outside of the domain of coexistence, enhanced costs, and benefits destabilize the system.

Impacts of plant natural enemies on a plant–fungi dynamics

Consistent with previous work, antagonist interactions (such as plant-enemy) can stabilize mutualism, by introducing neg- ative feedbacks (Holland et al. 2002). Analysis of the system with a fixed natural enemy population showed that if the costs of mycorrhizal associations to the fungi are high enough, then Fig. 5 Figures representing the effects of mycorrhizal fungi on the null an increase in natural enemy population could stabilize the isoclines of the prey–predator model system (Fig. 7, Appendix 2). However, the increase in natural Theor Ecol

Fig. 6 Model simulations for one plant species (X) in the presence of natural enemies (Y) and with fixed mycorrhizal fungal population (M)and parameters. Panels a and b illustrate two situations where mycorrhizal fungi, respectively, stabilize and destabilize the prey–predator dynamics enemy population needs to remain in the domain of coexis- as an increase in of the host (2004), rather tence defined in the analysis of the full system (Appendix 1). than an increase in growth rate as we did. It would be inter- If natural enemy population becomes too abundant, it will esting to expand upon our model to test if an increase in soil push plant population to extinction (Fig. 7). fertility would enhance parasitism. Furthermore, exogenous It is interesting to note that in the absence of the natural fluctuations in plant-enemy and mycorrhizal fungal feedbacks enemy population, the model represents a bidirectional con- might follow non-random patterns. For example, the accumu- sumer–resource interaction between a facultative host (plant) lation of pathogens over time might result in an increase in that undergo intraspecific competition and an obligate mutu- negative feedbacks from natural enemies (Klironomos 2002; alist (mycorrhizal fungi). Holland and DeAngelis (2010)stud- Diez et al. 2010). In contrast, host specificity in mycorrhizal ied a similar model in depth without competition, and they fungal might lead to an increase in positive feedbacks over showed that mutualism described in such a way was generally time (Kardol et al. 2006). Although important for understand- stable. Furthermore, they demonstrated that the range of dy- ing natural variation in community dynamics, such exogenous namics predicted by this type of mutualistic interactions was variation in enemy and mycorrhizal fungal effects would not similar to prey–predator interactions. In particular, cycles and change the results concerning plant species coexistence of our dampen oscillations towards the equilibrium were two possi- model as we included this level of variation in the mathemat- ble outcomes. Consistent with their findings, our simulations ical analyses (Appendix 1). led to similar dynamics (Fig. 7).

Effect of the environment on complex dynamic systems Conclusion An important consideration is that the effects of microbial feedbacks on plant community dynamics could be endoge- Our model considers the simultaneous effects of mycorrhizal nous (depend solely on the fungal–plant associations) or driv- fungi and negative feedbacks from enemies as mechanisms en by exogenous temporal variability in the physical environ- driving fluctuations in plant species abundance and fostering ment (e.g., precipitation or soil fertility). In particular, previ- species coexistence. Previous work examining the role of ous research suggests an increase in the costs of mycorrhizal plant–soil feedbacks in plant species coexistence concluded associations with increasing soil fertility and shade, with a that diversity could be maintained by negative but not positive potential shift from mutualistic to parasitic associations feedbacks (Bever et al. 1997). Other modeling efforts have (Neuhauser and Fargione 2004). In our model, mycorrhizal demonstrated that the joint effects of competition, mutualism, fungal associations result in costs and benefits for the plants and predation could lead to species coexistence (Neuhauser similarly to Neuhauser and Fargione (2004). However, and Fargione 2004; Holland et al. 2013; Georgelin and Nehauser and Fargione incorporated the benefits of mutualism Loeuille 2014; Mougi and Kondoh 2014). Our model Theor Ecol

Fig. 7 Model simulations for one plant species (X) in the presence of enemy population stabilizes a mutualism system that exhibits cycles (a), mycorrhizal fungi (M) and with fixed natural enemy population (Y)and that is stable (b), or that is unstable (c) parameters. Panels a, b,andc illustrate three situations where natural synthesizes these efforts by considering the joint effects of 1984;Freedmanetal.1987;Ringeletal.1996;Jang2002; mutualism, predation, and competition. Furthermore, we Bronstein et al. 2003;Meliánetal.2009; Loeuille 2010; added an explicit interaction between mutualism and preda- Mougi and Kondoh 2012; Holland et al. 2013; Georgelin tion. By combining these three important dynamics and their and Loeuille 2014; Mougi and Kondoh 2014). Early work in interactions, our approach is more ecologically realistic than this area typically includes only two species (May 1976; previous models. Christiansen and Fenchel 1977; Vandermeer and Boucher Specifically, our analysis addresses the effects of these 1978;Addicott1981). Subsequently, the focus shifted towards complex trophic dynamics on plant species coexistence, fur- three-species interactions in which a mutualist would arise ther developing the integration of mutualisms in food web and from a prey–predator or a competitor system (Rai et al. community dynamics (Rai et al. 1983; Addicott and Freedman 1983; Addicott and Freedman 1984; Freedman et al. 1987). Theor Ecol

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